The question is does it make noise when the a/c is off? If it doesn't but does when the ac is turned on it is causing the belt to slip which is hard to determine right now as the original poster has not responded to the thread.
The compressor causes a certain amount of drag when turned on as it takes energy to turn it once the clutch is engaged, that does not mean it has a bad bearing you start with the simple things first when trying to resolve problems.

thanks sir mal,  yes i caught it right away....

Moral of the lesson:  you have to be sensitive and pro-active about your mini.  if you hear squealing and clanking underneath the car, ..then search, investigate explore ,discover and fix...  it might be something serious and can cause accident,, ..i value my life more than anything else.

..and also sir mal, how and why would the belt slip IF the alternator pulley and the a/c pulleys are rotating as well with the engine?. that only means that if any of the alternator and a/c pulleys are not rotating as it should, then the bearing is shot, or something is hindering it from rotating.

If you are replying to my post for a person with the education you "claim" to have you certainly do not read and and absorb information. 
If you read my post i referred to a belt that had been slipping for a while where you seem to have implied you tightened yours right away which is why you were able to save the belt.

Maybe i will put it on a You tube video that you seem to like so much that way you can figure it out. 

my fan belt is 2 and a half years old , one time i noticed that it squealed , i tightened the alternator swing bracket then it stoped..., and God knows how often i drive my mini, when i get off my garage at 5am, i drive continuously for 3hrs and yet, my belt is good as ever.,no compound no nothing. i have my old one in the back of my car for spare.

If the AC is not on, and you accelerate, is the whistling noise still there?

From my observations once a Mini a/c or alternator belt slips and screeches long enough or any other belt that does the same to where it becomes annoying it is garbage and re tightening it if it helps will only alleviate the issue for a short time and may as Dan suggested do damage to other bearings.
Another issue i have run into with a/c equipped Mini's is the owner tries to tighten the belt with the adjuster but never loosened the idler pulley first which has to be done otherwise you are just over tightening the adjuster bolt to the point of stripping the threads, and i have seen a shop do the same thing.

(Specialist is an expert on useless "advices").

We did not discuss bearings (yet) because the symptoms reported were air conditioning that conditioning that works best on "full blast" and  "pressing on the gas creates this loud and obnoxious noise". Noisy/worn bearings would likely produce a constant sound.
The first thing to eliminate is the problem of a slipping belt. If it is squealing badly it is glazed (but probably not cracked) and just tightening it until the noise stops may damage bearings that very likely are currently in good condition.

you know mizzle,  those advice given to you by my colleagues are good, IF the problem is the belt, but since we do not know if the noise comes from the belt or bearing, those advices becomes useless. 
re-tighten the belt and observe if it still squealing.  Applying compound on the belt helps but only temporary, if the compound evaporated due to heat, then squeal comes back.  Examine the inside of the belt, if there are cracks then replace it, if there's none, then just re-tighten it. but still, you need to make sure that the noise is not cause by a shot alternator or compressor bearing. i'm surprise this was not mentioned by our 2 topnotch in house mechanic/ they focus their minds only to the belt.






get out of the box......and discover

To add to what Mur says, belts stretch over time, getting looser and beginning to slip, though maybe not noticeably. Slippage polishes the belt and the pulleys, reducing friction. A belt can also get glazed by moisture, dust and pollutants under the hood/bonnet. Simply increasing the tension until it stops squealing is not the right solution as the excess tension puts excessive load on the bearings of the water pump, alternator and power steering pump, if the car is so equipped.
The right thing to do is to adjust the belt to spec, and if that doesn't work, replace the belt, cleaning the pulleys as you go. Another short-term solution is to apply a belt dressing compound which may improve the friction between the belt and pulleys.

Do NOT listen to a word that Tony/Robster/Specialist has to say. He is a troll who has been banned repeatedly.

If the belt or belts have been squealing because of inadequate tension, they will be worn.

mizzle,
there are few possibilities:
1) the belt is loose and needs some tensioning/ belt is not necessarily worn out
2) A/C compressor bearing might be shot due to lack or total loss of lubrication.
3)Alternator bearing may also be shot. // wiggle it just a little bit and observe if there is play
4) do this:
  rev the car with a/c on--- can you hear the noise or no?
  rev the car with A/C off,,, ---can you hear the noise or no?
  drive the car with a/c on ---can you hear the noise or no?
  drive the car with a/c off,--- can you hear the noise or no?

this is how to diagnose a car without spending anything.


and nope,,, the famous ,well known specialist is 8,000 miles away from Japan.....

The belt or belts are worn AND out of adjustment. Since you are in Japan, there are mini specialists who can easily solve this issue. Whereabouts are you, perhaps you are near one of the well known mini specialists.

Good morning/evening Mini Community,
 
I am brand spanking new and I literally just purchased a 1995 Mini Cooper. My A/C seems to be working, but only when it's full blast. Since I'm in japan and the humidity is horrible, I was happy to find it worked. However, pressing on the gas creates this loud and obnoxious noise. and just FYI, I don't have a lot of mechanical background, so you might have to dumb it down for me.

Thanks!

#ILOVEMYMINI